Existence of ground state solutions to some Nonlinear Schr\"{o}dinger equations on lattice graphs

TL;DR

This paper proves the existence of ground state solutions for nonlinear Schrödinger equations on lattice and quasi-transitive graphs using variational methods, under certain conditions on the potential and nonlinearity.

Contribution

It extends the existence results of ground state solutions from lattice graphs to more general quasi-transitive graphs, using the Nehari method.

Findings

Existence of ground state solutions on lattice graphs under growth conditions.

Extension of results to quasi-transitive graphs.

Application of Nehari method to nonlinear Schrödinger equations.

Abstract

In this paper, we study the nonlinear Schr\"{o}dinger equation $ -\Delta u+V(x)u=f(x,u) $on the lattice graph $ \mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic or bounded, the above equation admits a ground state solution. Moreover, we extend our results from $\mathbb{Z}^{N}$ to quasi-transitive graphs.